This will generate a webpage of your completed tasks in text/csv, including task notes. It's not pretty, but it's really fast. I then use the browser's built in "Find on Page" feature (Ctrl-F) to search. The completion date is easy to identify because it is the last item in each line.

Of course, this is only good for Pro accounts, since the free ones only keep a week of history.

I have a Saved Search for completed tasks. The CTRL-F feature works (although I rarely use it), and when I find the task, I can actually do something with it, e.g. copy, edit, clone, delete, or "un-complete". Viewing the completion date requires sorting by "Date Completed", which takes a few extra seconds, but it's worth it to have direct access to the tasks.